It’s possible, but it’s not at all common, due to the facts that legitimizations rarely occur and that the bastards of nobleborn women (i.e., with house names) are rare. Bastards are rarely legitimized because you have to appeal to the king to do it, so it tends to be very exceptional. (We basically only have the examples of Aegon IV’s bastards, a few of the DoD’s dragonseeds, Ramsay Bolton, and maybe Jon.) And nobleborn women usually don’t have bastards because Westeros is so patriarchal — a woman would most likely have to be unmarried (otherwise she could claim it was her husband’s child, and even him being dead for a year wouldn’t prove otherwise) and have to go through with the pregnancy, and so on.
So in the books we have only a very few situations where a known nobleborn woman had a known bastard. In history, Elaena Targaryen had two bastard children by Alyn Velaryon, and there were Aegon IV’s Great Bastards by his nobleborn mistresses — but we only know them by bastard names, even the legitimized ones. In legend, there was Bael the Bard and Brandon the Daughterless’s daughter, whose bastard son became Lord Stark presumably because there was no other heir. And in modern times there’s only two situations that we know of — that is, Tyrion Tanner, son of Lollys Stokeworth; and Edric Storm, son of Delena Florent and Robert Baratheon.
(Ellaria Sand’s daughters don’t count since she’s a bastard herself, and Cersei’s children don’t count since they’re not known bastards. Similarly, if R+L=J but there was no marriage, Jon Snow doesn’t count (yet) since by all public accounts he’s Ned’s son. And there’s the whole complex Mormont women situation, where neither Maege nor Alysane admits their children are bastards despite them being unmarried — technically the girls and Alysane’s children should be Snows, but they’re Mormonts and nobody dares argue.)
Now, Edric Storm has the situation where if he were legitimized he would almost certainly take the name Baratheon, as it’s a Great House and also Robert was the king and he worships him. But as for Tyrion Tanner… well, he doesn’t have any known father, and his mother’s husband, Ser Bronn of the Blackwater, has no House name himself and so only goes by Lord Protector of Stokeworth. So if Tyrion were to be legitimized (which could happen if Bronn and Lollys have no legitimate children and Stokeworth needs an heir), he would almost certainly become Tyrion Stokeworth.
So you see, it’s very rare. It could happen… and I wouldn’t be surprised if it had happened more than once in Dorne, which is slightly less patriarchal due to their tradition of absolute primogeniture. But, basically, for your question, it would have to be a situation where an unmarried noblewoman has a child by an unknown or lower-ranked father, and for inheritance purposes that child is legitimized to take the mother’s name. Or possibly, if the father were higher-ranked but the child chooses to take their mother’s name for whatever reason. But I can only imagine that being an extremely rare exception.
I knew Alysane was unmarried and thus her children are Snows. However, maybe Maege has/had a husband we haven’t heard about. Assuming otherwise:
It seems Maege’s daughters are the only Mormont heirs. If someone inherits from someone with a different House name, don’t they change their name accordingly? In this case, Snow->Mormont; Maege’s daughters are just doing that a bit ahead of time. For another example, Harrold Hardyng would become Harrold Arryn if he succeeds Sweetrobin, right?